Finite Group Databasebeta

← #376402

Subgroups — #376402

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23730C211yesmaximal, characteristic
2#21043C211yesmaximal, characteristic, complemented
3#20886C211yesmaximal, characteristic
4#20446C211yesmaximal, characteristic, complemented
5#20422C211yesmaximal, characteristic
6#11364C211yesmaximal, characteristic, complemented
7#11212C211yesmaximal, characteristic, complemented
8#11126C211yesmaximal, characteristic
9#10959C211yesmaximal, characteristic
10#10944C211yesmaximal, characteristic
11#10910C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10893C211yesmaximal, characteristic, complemented
14#10871C211yesmaximal, characteristic, complemented
15#10751C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2905C2211yescharacteristic
22C2×#421C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#410C2211yescharacteristic
27#2873C2211yescharacteristic
28#2873C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2297C2211yescharacteristic
32#2285C2211yescharacteristic
33#2097C411yescharacteristic, complemented
34#2097C411yescharacteristic, complemented
35#2096C411yescharacteristic, complemented
36#2095C411yescharacteristic, complemented
37#2094C411yescharacteristic, complemented
38#2093C411yescharacteristic, complemented
39#2092C411yescharacteristic, complemented
40#2092C411yescharacteristic, complemented
41#1922—22no—
42#1922—22no—
43#1866—22no—
44#1866—22no—
45#1773—22no—
46#1772—22no—
47#1765—22no—
48#1764—22no—
49#1740C2211yescharacteristic
50#1740C2211yescharacteristic
51#1740C2211yescharacteristic
52C2×#341C2211yescharacteristic
53#1735C2211yescharacteristic
54#1733C2211yescharacteristic
55#1733C2211yescharacteristic
56#1733C2211yescharacteristic
57#1733C2211yescharacteristic
58#1733C2211yescharacteristic
59#1733C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C2×#339C2211yescharacteristic
64#1729C2211yescharacteristic
65#1729C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68#568—22no—
69#568—22no—
70#568—22no—
71#568—41no—
72C2×#131—22no—
73C2×#131—22no—
74C22×M16—41no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#108—22no—
90C2×#108—22no—
91C2×#108D412yes—
92C2×#108D412yes—
93#427—21no—
94#427—21no—
95#427—21no—
96#427—21no—
97#427—21no—
98#427—21no—
99#427—21no—
100#427—21no—
101#427—22no—
102#427—41no—
103C2×#107—21no—
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#106—21no—
109C2×#106C2×C411yescharacteristic
110C2×#106C2×C411yescharacteristic
111C2×#106C2×C411yescharacteristic
112C2×#106C2×C411yescharacteristic
113#424—22no—
114#424—22no—
115#423—41no—
116#421—22no—
117#421—22no—
118#421—41no—
119#421Q814yes—
120C2×#104—22no—
121C2×#104—22no—
122#416—21no—
123#416—21no—
124#416—21no—
125#416—21no—
126#416—21no—
127#416—21no—
128#416—21no—
129#416—21no—
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#102C2×C411yescharacteristic
138#410—22no—
139#410—22no—
140C2×#99—41no—
141#405—22no—
142#405—22no—
143#405—22no—
144#405—22no—
145#405—22no—
146#405—22no—
147#405—22no—
148#405—41no—
149C4×M16—41no—
150C4×M16—41no—
151C2×#97—21no—
152C2×#97—21no—
153C2×C4×C8—21no—
154C2×C4×C8—21no—
155#391C2×C411yescharacteristic
156#391C2×C411yescharacteristic
157#390C2×C411yescharacteristic
158#390C2×C411yescharacteristic
159#386—41no—
160#384C2×C411yescharacteristic
161#384C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#382C2×C411yescharacteristic
164#380—41no—
165C4×C4⋊C4—41no—
166C4×C22⋊C4—41no—
167#342—41no—
168#342—41no—
169#342—41no—
170#341D412yes—
171#341D412yes—
172#340—21no—
173#340—21no—
174#339—22no—
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#339—22no—
180#339—22no—
181#339—22no—
182#339—41no—
183#339—41no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#336C2×C411yescharacteristic
189#336C2×C411yescharacteristic
190#336C2×C411yescharacteristic
191#336C2×C411yescharacteristic
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41C2×Q812yes—
203C2×#41#4112yes—
204C22×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C2411yescharacteristic
210#131—28no—
211#131—44no—
212#131—44no—
213#131—44no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—42no—
231C2×M16—44no—
232C2×M16—44no—
233C2×M16C2×D412yes—
234C2×M16C2×D412yes—
235C2×M16C2×D412yes—
236C2×M16C2×D412yes—
237C2×M16C2×Q812yes—
238C2×M16#4112yes—
239C2×M16#4112yes—
240C2×M16#4112yes—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—42no—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C411yescharacteristic
257C22×C8C22×C411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8#4111yescharacteristic
263C22×C8#4111yescharacteristic
264C22×C8#4111yescharacteristic
265C22×C8#4111yescharacteristic
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117C2×Q812yes—
293#117#4112yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329#108—24no—
330#108—24no—
331#107—21no—
332#107—21no—
333#107—21no—
334#107—21no—
335#107—21no—
336#107—21no—
337#107—21no—
338#107—21no—
339#107—42no—
340#106—21no—
341#106—21no—
342#106—21no—
343#106—21no—
344#106—21no—
345#106—21no—
346#106—21no—
347#106—21no—
348#106—42no—
349C4⋊C8—44no—
350#104—28no—
351#104—44no—
352#103—21no—
353#103—21no—
354#103—21no—
355#103—21no—
356#103—21no—
357#103—21no—
358#103—21no—
359#103—21no—
360#102—21no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#102—21no—
366#102—21no—
367#102—21no—
368#99—44no—
369#97—24no—
370#97—42no—
371#97—42no—
372#97—42no—
373C4×C8—24no—
374C4×C8—42no—
375C4×C8—42no—
376C4×C8—42no—
377#95—41no—
378#95—41no—
379#95—41no—
380#95—41no—
381C24—21no—
382C24H(5,𝔽2)11yescharacteristic
383#41—22no—
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8C4×Q812yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4C4×Q812yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—81no—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×Q811yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C42−1+411yescharacteristic
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—48no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16C22⋊Q814yes—
470M16#12314yes—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—22no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—48no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8C4×D411yescharacteristic
509C2×C8C4×D411yescharacteristic
510C2×C8C4×D411yescharacteristic
511C2×C8C4×D411yescharacteristic
512C2×C8C4×D411yescharacteristic
513C2×C8C4×D411yescharacteristic
514C2×C8C4×D411yescharacteristic
515C2×C8C4×D411yescharacteristic
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—22no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—81no—
580C23—81no—
581C23#51811yescharacteristic
582C23#54011yescharacteristic
583C23#54211yescharacteristic
584C23#55011yescharacteristic
585C23#55011yescharacteristic
586C23#55111yescharacteristic
587C23#55411yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592D4—22no—
593D4—22no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—41no—
599D4—41no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×Q811yescharacteristic
642C2×C4C4×#4111yescharacteristic
643C2×C4C4×#4111yescharacteristic
644C2×C4#52011yescharacteristic
645C2×C4C2×C22⋊Q811yescharacteristic
646C2×C4C2×#12311yescharacteristic
647C2×C4#52511yescharacteristic
648C2×C4#52511yescharacteristic
649C2×C4#52911yescharacteristic
650C2×C4#52911yescharacteristic
651C2×C4#52911yescharacteristic
652C2×C4#53311yescharacteristic
653C2×C4#53811yescharacteristic
654C2×C4#53911yescharacteristic
655C2×C4D4211yescharacteristic
656C2×C4#54611yescharacteristic
657C2×C4#54611yescharacteristic
658C2×C4#54711yescharacteristic
659C2×C4D4×Q811yescharacteristic
660C2×C4#55111yescharacteristic
661C2×C4#55111yescharacteristic
662C2×C4#55211yescharacteristic
663C2×C4#55211yescharacteristic
664C2×C4#55411yescharacteristic
665C2×C4#55411yescharacteristic
666C2×C4#55911yescharacteristic
667C8—21no—
668C8—21no—
669C8—21no—
670C8—21no—
671C8—24no—
672C8—24no—
673C8—21no—
674C8—21no—
675C8—21no—
676C8—21no—
677C8—42no—
678C8—42no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—42no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22—161no—
694C22#233511yescharacteristic
695C22#236611yescharacteristic
696C22#236811yescharacteristic
697C22#242311yescharacteristic
698C22#245411yescharacteristic
699C22#247111yescharacteristic
700C22#247911yescharacteristic
701C4—42no—
702C4—42no—
703C4—81no—
704C4—82no—
705C4—82no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—161no—
713C4—161no—
714C4C4×#12311yescharacteristic
715C4C4×C22⋊Q811yescharacteristic
716C4#235911yescharacteristic
717C4#236011yescharacteristic
718C4#241811yescharacteristic
719C4#244911yescharacteristic
720C4#257511yescharacteristic
721C4#258911yescharacteristic
722C2—21no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—82no—
729C2#1417811yescharacteristic
730C2#2319611yescharacteristic
731C2#2319711yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2366
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21043C2semidirect
4. #20446C2semidirect
6. #11364C2semidirect
7. #11212C2semidirect
11. #10910C2semidirect
12. #10909C2semidirect
13. #10893C2semidirect
14. #10871C2semidirect
33. #2097C4semidirect
34. #2097C4semidirect
35. #2096C4semidirect
36. #2095C4semidirect
37. #2094C4semidirect
38. #2093C4semidirect
39. #2092C4semidirect
40. #2092C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237302
#210432
#208862
#204462
#204222
#113642
#112122
#111262
#109592
#109442
#109102
#109092
#108932
#108712
#107512